Every group admits an initial homomorphism to a pi-powered group

Statement

Suppose $G$ is a group and $\pi$ is a set of primes. There exists a $\pi$-powered group $K$ and a homomorphism of groups $\alpha:G \to K$ such that for any $\pi$-powered group $L$ and any homomorphism $\varphi:G \to L$, there is a unique homomorphism $\theta:K \to L$ such that $\varphi = \theta \circ \alpha$.

We can think of the functor that sends $G$ to the group $K$ as the free $\pi$-powering functor. It is the left-adjoint functor to the forgetful functor from $\pi$-powered groups to groups.

In the localization terminology

The term $\pi$-local is sometimes used to refer to a group that is powered over all the primes not in $\pi$. The functor described above is, in that context, termed the $\pi'$-localization functor. By $\pi'$ we mean the complement of $\pi$ in the set of all primes.

Related facts

• The free powered group for a set of primes can be thought of as being obtained from the abstract free group by applying the free $\pi$-powering functor.

Embeddability results

We are interested in cases where the canonical homomorphism from the group to its $\pi$-powering is injective, and in related questions, some of which are explored below.

Group assumption on both the starting group and the big group Divisibility/powering/torsion assumption on the starting group Divisibility/powering/torsion assumption on the big group Is this always possible? Proof
nilpotent group none divisible group No nilpotent group need not be embeddable in a divisible nilpotent group
nilpotent group $\pi$-torsion-free group (equivalently, $\pi$-powering-injective group) $\pi$-powered group Yes every pi-torsion-free nilpotent group can be embedded in a unique minimal pi-powered nilpotent group
arbitrary group none divisible group Yes every group is a subgroup of a divisible group, every pi-group is a subgroup of a divisible pi-group
arbitrary group $\pi$-torsion-free group (equivalently, $\pi$-powering-injective group) $\pi$-powered group No Powering-injective group need not be embeddable in a rationally powered group